A First Approach to Numerical Sets: From Naturals to Complex
Summary:In this class, we will explore how natural numbers can be used as the basis for constructing other numerical sets to overcome certain operational limitations. We will start with the integers, which allow us to carry out subtractions extensively. Then, we will move on to rational numbers, which provide us with the tool of division in a complete way. Subsequently, we will delve into real numbers to be able to work with n-th roots, and we will mention how complex numbers are introduced to address specific scenarios with n-th roots. Through these developments, it will be understood how each new numerical set arises to solve problems inherent to the previous one.
Learning Objectives:
Upon completing this class, the student will be able to:
- Identify the basic properties of natural numbers, integers, and rational numbers.
- Interpret the properties and basic operations that are inherited or modified when transitioning from one numerical set to another.
- Compare the properties of different numerical sets and how they relate to each other.
TABLE OF CONTENTS
Introduction
Properties of Natural Numbers
Transition from Natural Numbers to Integers
The Leap to Rational Numbers
Real and Irrational Numbers
The Complex: The Algebraic Clause of Real Numbers
Introduction
The real numbers, along with other numerical sets that we will explore in this class, are introduced by expanding natural numbers. It happens that, with any two natural numbers, it is not always possible to perform subtraction or division operations, and these expansions aim to solve this inconvenience.
During this class, we will review the operations and properties of natural numbers, and on this basis, we will move towards the construction of all other numerical sets, until reaching real numbers and beyond.
Properties of Natural Numbers
When addressing operations with natural numbers, we mainly refer to addition and multiplication, along with their respective inverse operations. The following summarizes these properties:
Given that a,b,c\in\mathbb{N}, it is verified that:
| 1. | a + b = b + a |
| 2. | a \pm (b \pm c) = (a\pm b)\pm c (in the case of subtraction, it is valid as long as it is well-defined) |
| 3. | a\cdot b = b \cdot a |
| 4. | a\cdot(b\cdot c)= (a\cdot b)\cdot c |
| 5.\;\;\;\;\; | a\cdot b = a \leftrightarrow b=1 |
| 6. | \displaystyle \frac{a}{b}\in\mathbb{N} \leftrightarrow (\exists k\in\mathbb{N})(a=b\cdot k) |
| 7. | a\cdot(b+c)=a\cdot b + a \cdot c |
Transition from Natural Numbers to Integers
The first aspect to note is that in the case of sums: (\forall a,b\in\mathbb{N})(a+b\in\mathbb{N}), while for subtractions: (\forall a,b\in\mathbb{N})(a+b\in\mathbb{N} \leftrightarrow a\gt b). An inconvenience arises when the subtraction between two natural numbers a and b does not make sense if a\leq b; to remedy this situation, natural numbers are expanded to the set of integers, where subtractions of this nature acquire a well-defined value. We denote this new set of integers with the letter \mathbb{Z}, and it consists of all natural numbers, their additive inverses, and zero.
\mathbb{Z} = \{\cdots, -3,-2,-1,0,1,2,3,\cdots \}
Integers inherit all the properties and operations of natural numbers, with an extension on the second property, and the notions of inverse and additive neutral are introduced.
| 2*. | a \pm (b \pm c) = (a\pm b) \pm c |
| 8. | (\forall a\in\mathbb{Z})(\exists ! b\in\mathbb{Z})(a+b=0 \leftrightarrow b=-a) |
| 9. | (\forall a\in\mathbb{Z})(\exists ! b\in\mathbb{Z})(a+b=a \leftrightarrow b=0) |
The element b=-a is what we call the additive inverse of a.
The Leap to Rational Numbers
At this point the only operation that remains undefined is division. To solve this we will expand upon the set of integers to the set of rational numbers, which will be given by the following set:
\mathbb{Q}=\left\{a= \displaystyle\frac{n}{m}\;|\;n,m\in\mathbb{Z}\wedge m\neq 0 \right\}
This acquires a new property
| 10. | (\forall a \in \mathbb{Q}\setminus\{0\})(\exists ! b \in \mathbb{Q}) \left[(a\cdot b = 1) \leftrightarrow \left( b = \displaystyle \frac{1}{a} = a^{-1} \right)\right] |
| Every non-zero rational has a multiplicative inverse. The multiplicative inverse of a is a^{-1} | |
With these numbers, operations and properties, new operations with their properties are defined. In these, the nth power of a rational q is defined through
q^n = \underbrace{q\cdot q \cdot \cdots \cdot q}_{n\;veces}; with n\in\mathbb{N}
q^{-n}= \displaystyle \frac{1}{q^n}
Note that, from this, and whenever q\neq 0, we can say that
q^0 = 1
Moreover, whenever zero divisions appear, given any two rationals a,b , and two integers n,m the following properties will be fulfilled:
| 11. | a^n \cdot a^m = a^{n+m} |
| 12. | (a^n)^m = a^{n\cdot m} |
| 13. | (a\cdot b)^n = a^{n} \cdot a^{m} |
| 14. | \left(\displaystyle \frac{a}{a}\right)^n = \frac{a^n}{a^n} |
| 15. | \displaystyle \frac{a^n}{a^m} = a^{n-m} = \frac{1}{a^{m-n}} |
Real and Irrational Numbers
Just as the operation of subtraction (inverse of addition) and division (inverse of multiplication) made it necessary to expand the natural numbers to integers and rationals, respectively, to form well-defined operations, similarly occurs with powers. The inverse operation of the n-th power is the n-th root.
Definition of Root
Let n be an integer greater than 1 and p,q any rational numbers, the n-th root of q is defined, which we represent through the following rules:
| 16. | q=0 \rightarrow \sqrt[n]{q} = 0 |
| 17. | q \gt 0 \rightarrow \left[ \sqrt[n]{q} = p \leftrightarrow p^n = q \right] |
| 18. | \left[ q \lt 0 \wedge n {\;is\;odd} \right]\rightarrow \left[ \sqrt[n]{q} = p \leftrightarrow p^n = q \right] |
In summary, the n-th root of q is a number p such that, when raised to n, it gives you back the number q. In these cases, when n=2, instead of writing \sqrt[2]{q}, for simplicity we write \sqrt{q}.
The Emergence of Irrational Numbers
At this point, we wonder is the n-th root well-defined for all elements of \mathbb{Q}? The truth is that, although not so evident (compared to what we saw with subtraction and division), there are rationals that do not have a rational n-th root. To see this, it is enough to review the following example:
\sqrt{2} is not a rational number.
PROOF
We will prove this by reduction to the absurd.
Suppose that \sqrt{2} is a rational number, that is, there are p,q\in\mathbb{Z}, with q\neq 0, such that \sqrt{2}=p/q, and that it has also been simplified until becoming irreducible. If we do so then we can say that
2 = \left(\sqrt{2} \right)^2 =\displaystyle \frac{p^2}{q^2} = \left(\displaystyle \frac{p}{q}\right)^2
But this contradicts the fact that p/q was written in irreducible form (now it turns out that (p/q)^2 can be simplified and its result is 2). Since assuming that \sqrt{2} is rational produces a contradiction, then it cannot be a rational number and we say, consequently, that it is irrational.
The Expansion to Real Numbers
These results highlight the fact that, to correctly define the n-th root it is necessary to extend the rationals to a new set, this is the set of real numbers, which we denote by \mathbb{R} and that contains both rationals and irrationals
\mathbb{R}= \mathbb{Q}\cup \mathbb{Q}^*
The Complex: The Algebraic Closure of Real Numbers
At this point, we should note two things: (1) when n is even, the n-th root becomes multivalued and, (2) if we also try to calculate \sqrt[n]{q} with q\lt 0, we will see that such a number cannot be a real number.
The first is resolved by defining the principal root by applying a slight change to point (17) that talks about the definition of the root, staying as follows:
| 17*. | q\gt 0 \rightarrow \left[ 0\lt p=\sqrt[n]{q} \leftrightarrow p^n=q \right] |
The second is achieved by extending the set of real numbers to the set of complex numbers \mathbb{C}, but this construction will be discussed later on.